Anything with state comes to mind. The text field I am typing this into, for example. I type on my keyboard and the state of the text field changes, and after I hit "reply", the state of the page changes with a comment appended. Before/after.
Yes, you can implement this by creating a completely new page containing the concatenation of the old page with the comment, but externally (when you visit the URL), the state of that page has changed. So if you choose to implement the problem in a functional style, you have to bridge that gap somehow between that style and the problem at hand.
Any sort of document processing done on computers in Word, Excel (regarding the document itself, not the one way dataflow constraint program inside), OpenOffice, PowerPoint, Pages, Keynote, Quark XPress, InDesign, Photoshop, Illustrator etc. People use these programs to change the state of documents. That is the purpose of these programs.
Anything that interacts with the world, for example the user interface.
Or Wikipedia. Pages change, sometimes because there is new information, sometimes because something in the world has changed. Or most any other web site.
Really, the world (a) has (a lot of) state and (b) that state is changing incessantly. It is not (a) stateless or (b) immutable.
But don't take it from me: "In the end, any program must manipulate state. If it doesn't, there is no point in running it. You hit go and the box gets hotter" - Simon Peyton-Jones. https://www.youtube.com/watch?v=iSmkqocn0oQ&t=3m20s